Sampling Distribution

What is a Sampling Distribution?

A sampling distribution can be defined as a probability distribution using statistics by first choosing a particular population and then making use of random samples which are drawn from the population i.e. it basically targets at the spreading of the frequencies related to the spread of various outcomes or results which can possibly take place for the particular chosen population.

Explanation

A lot of researchers, academicians, market strategists, etc. go ahead of sampling distribution instead of choosing the entire population. This makes the data set easy and also manageable. To make it easier suppose a marketer wants to do an analysis of the number of youth riding a bicycle between two regions within the age limit 13-18.

For this purpose, he will not take into account the entire population present in the two regions between 13-18 years of age which is practically not possible and even if done it too time-consuming and data set is not manageable. Instead, the marketer will take a sample set of 200 each from each region and get the sampling distribution done.

The average count of the usage of the bicycle here is termed as the sample mean. Each sample chosen has its own mean generated and the distribution done for the average mean obtained is defined as the sample distribution. The deviation obtained is termed as the standard error.

How to Calculate Sampling Distribution?

This primarily involves the calculation of the mean and the associated standard error. According to the theorem of central limits, it states that when the sample size is very large, it will close to be equivalent to the population itself. When the distribution of the population is unaware to the researcher it is assumed to be a normal one.

Thus, the standard deviation is very important to estimate the sampling distribution. The mean can be obtained by adding up all the noted observations and dividing the same by the number of samples or observations. After this, we need to add 1/population size and 1/sample size. When the population size is huge taking just 1/sample size will also do.

The final step is taking the results of the second step mentioned above and finding the square root of it. The square root is then multiplied with the standard deviation. This gives us the measure of standard error. Combining both the standard error and the means help us estimate the sampling distribution.

Example of Sampling Distribution

Assuming that a researcher is conducting a study on the weights of the inhabitants of a particular town and he has five observations or samples i.e. 70kg, 75kg, 85kg, 80kg, and 65kg. The town is generally considered to be having a normal distribution and maintains a standard deviation of 5kg in the aspect of weight measures. Thus the mean can be calculated as (70+75+85+80+65)/5 = 75 kg.

Also, we assume that the population size is huge thus to go to the second step we will divide the number of observations or samples by 1 i.e. 1/5 = 0.20. Now we need to take the square root of 0.20 which comes to 0.45. The square root is then multiplied by the standard deviation i.e. 0.45*5 = 2.25kg. Thus standard error obtained is 2.25kg and the mean obtained was 75kg. These two factors can be used to describe the sampling distribution.

Types of Sampling Distribution

#1 – Sampling Distribution of Mean

This can be defined as the probabilistic spread of all the means of samples chosen on a random basis of a fixed size from a particular population. When samples have opted from a normal population the spread of the mean obtained will also be normal to the mean and the standard deviation.

If the population is not normal to still the distribution of the means will tend to become closer to the normal distribution provided that the sample size is quite large.

#2 – Sampling Distribution of Proportion

This is primarily associated with the statistics involved in attributes. Here the role of binomial distribution comes into play. Generally, it responds to the laws of binomial distribution but as the sample size increases, it usually becomes normal distribution again.

#3 – Student’s T-Distribution

This type of distribution is used when the standard deviation of the population is unknown to the researcher or when the size of the sample is very small. This type of distribution is very symmetrical and fulfills the condition of standard normal variate. As the sample size increases, even T distribution tends to become very close to normal distribution.

#4 – F Distribution

When the greater variance is mandatorily present in the numerator the F distribution finds its usage. As the degree of freedom changes the critical values of F changes too which is applicable for both large and small variances. This can be calculated from the tables available.

The comparison is made from the measured value of F belonging to the sample set and the value which is calculated from the table. If the earlier one is equal to or larger than the table value the null hypothesis of the study gets rejected.

#5 – Chi-Square Formula Distribution

This type of distribution is used when the data set involves dealing with values that include adding up the squares. The set of squared quantities belonging to the variance of samples is added and thus a distribution spread is made which we call as chi-square distribution.

Importance

This is important because it simplifies the path to statistical inference. Moreover, it allows analytical considerations to be focussed upon a static distribution rather than the mixed probabilistic spread of each chosen sample unit.

Elimination of variability present in the statistic is done by using this distribution.

It provides us with an answer about the probable outcomes which are most likely to happen.

They play a key role in inferential statistical studies which means they play a major role in making inferences regarding the entire population.

Conclusion

This is key in statistics because they act as a major guideline to statistical inference. They basically guide the researcher, academicians or statisticians about the spread of the frequencies signaling a range of varied probable outcomes that could be further tagged to the entire population.

The prime factor involved here is the mean of the sample and the standard error which if estimates help us calculate the sampling distribution too. There are various types of distribution techniques and based on the scenario and data set each is applied.

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